An example on the Mann iteration method for Lipschitz pseudocontractions
HTML articles powered by AMS MathViewer
- by C. E. Chidume and S. A. Mutangadura PDF
- Proc. Amer. Math. Soc. 129 (2001), 2359-2363 Request permission
Abstract:
An example of a Lipschitz pseudocontractive map with a unique fixed point is constructed for which the Mann iteration sequence fails to converge. This resolves a long standing open problem.References
- Felix E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044. MR 187120, DOI 10.1073/pnas.54.4.1041
- David Borwein and Jonathan Borwein, Fixed point iterations for real functions, J. Math. Anal. Appl. 157 (1991), no. 1, 112–126. MR 1109446, DOI 10.1016/0022-247X(91)90139-Q
- Shih Sen Chang, The Mann and Ishikawa iterative approximation of solutions to variational inclusions with accretive type mappings, Comput. Math. Appl. 37 (1999), no. 9, 17–24. MR 1687999, DOI 10.1016/S0898-1221(99)00109-1
- C. E. Chidume, Global iteration schemes for strongly pseudo-contractive maps, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2641–2649. MR 1451791, DOI 10.1090/S0002-9939-98-04322-6
- Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258 (German). MR 190718, DOI 10.1002/mana.19650300312
- Troy L. Hicks and John D. Kubicek, On the Mann iteration process in a Hilbert space, J. Math. Anal. Appl. 59 (1977), no. 3, 498–504. MR 513062, DOI 10.1016/0022-247X(77)90076-2
- Shiro Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150. MR 336469, DOI 10.1090/S0002-9939-1974-0336469-5
- W. A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965), 1004–1006. MR 189009, DOI 10.2307/2313345
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Qi Hou Liu, The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings, J. Math. Anal. Appl. 148 (1990), no. 1, 55–62. MR 1052044, DOI 10.1016/0022-247X(90)90027-D
- Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Additional Information
- C. E. Chidume
- Affiliation: The Abdus Salam International Centre for Theoretical Physics, 34100 Trieste, Italy
- MR Author ID: 232629
- Email: chidume@ictp.trieste.it
- S. A. Mutangadura
- Affiliation: Department of Physics, University of Zimbabwe, Harare, Zimbabwe
- Received by editor(s): December 14, 1999
- Published electronically: January 18, 2001
- Communicated by: Jonathan M. Borwein
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2359-2363
- MSC (2000): Primary 47H09, 47J25
- DOI: https://doi.org/10.1090/S0002-9939-01-06009-9
- MathSciNet review: 1823919